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‎Book4_Ch11_Python_Codes/Bk4_Ch11_01.ipynb

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‎Book4_Ch11_Python_Codes/Bk4_Ch11_02.ipynb

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‎Book4_Ch12_Python_Codes/Bk4_Ch12_01.ipynb

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‎Book4_Ch13_Python_Codes/Bk4_Ch13_01.ipynb

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‎Book4_Ch13_Python_Codes/Bk4_Ch13_02.ipynb

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‎Book4_Ch13_Python_Codes/Streamlit_Bk4_Ch13_04.py

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###############
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# Authored by Weisheng Jiang
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# Book 4 | From Basic Arithmetic to Machine Learning
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# Published and copyrighted by Tsinghua University Press
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# Beijing, China, 2022
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###############
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import streamlit as st # 导入 Streamlit 库，用于创建交互式 Web 应用
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import numpy as np # 导入 NumPy 库，用于数值计算
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import plotly.express as px # 导入 Plotly Express 库，用于绘制交互式图表
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import pandas as pd # 导入 Pandas 库，用于数据处理
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# 定义函数 bmatrix，用于将 NumPy 数组转化为 LaTeX 矩阵格式
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def bmatrix(a):
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"""返回一个 LaTeX 矩阵表示"""
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if len(a.shape) > 2: # 检查输入的数组是否为二维
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raise ValueError('bmatrix 函数最多显示二维矩阵') # 如果不是二维数组，抛出异常
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lines = str(a).replace('[', '').replace(']', '').splitlines() # 去掉数组的方括号并按行拆分
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rv = [r'\begin{bmatrix}'] # 开始 LaTeX 矩阵的表示
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rv += [' ' + ' & '.join(l.split()) + r'\\' for l in lines] # 将每一行的元素用 LaTeX 格式化
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rv += [r'\end{bmatrix}'] # 结束 LaTeX 矩阵的表示
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return '\n'.join(rv) # 返回拼接后的 LaTeX 字符串
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# 在侧边栏创建交互式滑块，用户可调整矩阵 A 的元素
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with st.sidebar:
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# 在侧边栏中展示一个 LaTeX 格式的矩阵模板
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st.latex(r'''
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A = \begin{bmatrix}
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a & b\\
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c & d
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\end{bmatrix}''')
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# 为矩阵 A 的元素 a, b, c, d 创建滑块，用户可调整这些值
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a = st.slider('a', -2.0, 2.0, step=0.1, value=1.0) # 滑块用于设置 a 的值，默认值为 1.0
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b = st.slider('b', -2.0, 2.0, step=0.1, value=0.0) # 滑块用于设置 b 的值，默认值为 0.0
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c = st.slider('c', -2.0, 2.0, step=0.1, value=0.0) # 滑块用于设置 c 的值，默认值为 0.0
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d = st.slider('d', -2.0, 2.0, step=0.1, value=1.0) # 滑块用于设置 d 的值，默认值为 1.0
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#%% 创建网格点用于二维平面上的点
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x1_ = np.linspace(-1, 1, 11) # 在 [-1, 1] 区间内生成 11 个均匀分布的点，用于 x1
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x2_ = np.linspace(-1, 1, 11) # 在 [-1, 1] 区间内生成 11 个均匀分布的点，用于 x2
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xx1, xx2 = np.meshgrid(x1_, x2_) # 创建二维网格，用于生成所有点的坐标
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X = np.column_stack((xx1.flatten(), xx2.flatten())) # 将网格点展开为二维数组，每行一个点的坐标
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# 定义矩阵 A，由用户调整的滑块值确定
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A = np.array([[a, b], # 矩阵 A 的第一行
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[c, d]]) # 矩阵 A 的第二行
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X = X @ A # 使用矩阵乘法，将点集 X 通过矩阵 A 进行线性变换
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#%% 创建颜色数组并将其添加到点数据中
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color_array = np.linspace(0, 1, len(X)) # 为每个点生成一个对应的颜色值，范围为 [0, 1]
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X = np.column_stack((X, color_array)) # 将颜色值添加到点数据中，作为第三列
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df = pd.DataFrame(X, columns=['z1', 'z2', 'color']) # 将点数据转换为 DataFrame，并命名列为 z1, z2, 和 color
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#%% 绘制散点图
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st.latex('A = ' + bmatrix(A)) # 在页面上以 LaTeX 格式展示矩阵 A
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# 使用 Plotly Express 绘制散点图
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fig = px.scatter(df, # 数据来源为 DataFrame
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x="z1", # z1 作为横轴
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y="z2", # z2 作为纵轴
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color='color', # 根据 color 列设置点的颜色
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color_continuous_scale='rainbow') # 使用彩虹色带表示颜色
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# 设置图形的布局参数
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fig.update_layout(
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autosize=False, # 禁用自动尺寸调整
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width=500, # 设置图形宽度为 500 像素
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height=500) # 设置图形高度为 500 像素
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# 添加横轴和纵轴的黑色参考线
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fig.add_hline(y=0, line_color='black') # 添加横轴参考线
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fig.add_vline(x=0, line_color='black') # 添加纵轴参考线
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# 设置坐标轴的显示范围
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fig.update_xaxes(range=[-3, 3]) # 设置 x 轴范围为 [-3, 3]
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fig.update_yaxes(range=[-3, 3]) # 设置 y 轴范围为 [-3, 3]
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# 禁用颜色条显示
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fig.update_coloraxes(showscale=False) # 隐藏颜色条
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# 在 Streamlit 页面中展示绘制的散点图
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st.plotly_chart(fig)
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‎Book4_Ch14_Python_Codes/Bk4_Ch14_01.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "73bd968b-d970-4a05-94ef-4e7abf990827",
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"metadata": {},
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"source": [
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"Chapter 14\n",
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"\n",
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"# 矩阵平方根\n",
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"Book_4《矩阵力量》 | 鸢尾花书：从加减乘除到机器学习 (第二版)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c263fc95-881a-49fb-9c6c-a25508996623",
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"metadata": {},
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"source": [
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"该代码的主要任务是通过矩阵分解和重构方法，基于给定的矩阵$A$，验证矩阵重构的正确性。具体流程如下：\n",
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"\n",
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"1. **定义矩阵$A$**： \n",
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" 代码首先创建了一个$2 \\times 2$矩阵 $A = \\begin{bmatrix} 1.25 & -0.75 \\\\ -0.75 & 1.25 \\end{bmatrix}$。矩阵 $A$ 是一个对称矩阵，因此可以进行特征值分解。\n",
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"\n",
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"2. **计算特征值和特征向量**： \n",
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" 代码使用 `np.linalg.eig` 函数对矩阵 $A$ 进行特征值分解，计算出$A$的特征值（存储在 $LAMBDA$ 中）和特征向量（存储在 $V$ 中），满足以下分解公式：\n",
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" $$\n",
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" A = V \\Lambda V^{-1}\n",
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" $$\n",
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" 其中，$\\Lambda$ 是一个包含特征值的对角矩阵，$V$ 是由特征向量组成的矩阵。\n",
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"\n",
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"3. **构建矩阵$B$**： \n",
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" 接下来，代码构建了一个新的矩阵 $B$。它的计算公式为：\n",
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" $$\n",
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" B = V \\sqrt{\\Lambda} V^{-1}\n",
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" $$\n",
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" 其中，$\\sqrt{\\Lambda}$ 是对角矩阵，其对角元素是 $\\Lambda$ 的平方根。也就是说，$B$ 是通过将 $A$ 的特征值取平方根后重新组合得到的矩阵。\n",
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"\n",
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"4. **重构矩阵$A$并验证结果**： \n",
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" 最后，通过矩阵 $B$ 构造了一个新矩阵 $A_{reproduced}$，其计算公式为：\n",
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" $$\n",
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" A_{reproduced} = B B^T\n",
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" $$\n",
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" 这是利用矩阵 $B$ 重构 $A$ 的过程。对称矩阵 $A$ 的特征值平方根分解使得 $B B^T$ 应等于原始矩阵 $A$。因此，通过打印 $A_{reproduced}$，可以验证 $B B^T$ 是否等于 $A$，从而确认重构的正确性。"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "2759881c-9e2a-4e4d-a79c-1617f2a4be4f",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np # 导入NumPy库"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a685835a-bcda-40f8-ac57-ff3738c0209f",
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"metadata": {},
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"source": [
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"## 初始化矩阵A"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "a6e9a15c-1d77-4a95-a13c-1c825598e8be",
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"metadata": {},
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"outputs": [],
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"source": [
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"A = np.matrix([[1.25, -0.75], # 定义矩阵A\n",
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" [-0.75, 1.25]]) # 矩阵的元素"
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]
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},
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{
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"cell_type": "markdown",
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"id": "55297e89-757e-4136-a0af-4763d1db3e56",
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"metadata": {},
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"source": [
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"## 计算特征值和特征向量"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "998fa920-5e90-408f-8b70-dd3633c870e9",
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"metadata": {},
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"outputs": [],
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"source": [
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"LAMBDA, V = np.linalg.eig(A) # 计算矩阵A的特征值（LAMBDA）和特征向量（V）"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "ce8e4f8c-19a0-4392-b1ef-e794dcc5f187",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([2. , 0.5])"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"LAMBDA"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "13196da8-54c0-44bb-ac29-6cf7c6fec408",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"matrix([[ 0.70710678, 0.70710678],\n",
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" [-0.70710678, 0.70710678]])"
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]
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},
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"V"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c6bb1a72-57de-4ae9-a2e2-599db3122538",
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"metadata": {},
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"source": [
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"## 构建矩阵B"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "cb92d4fe-8443-473f-a61c-378630468024",
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"metadata": {},
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"outputs": [],
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"source": [
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"B = V @ np.diag(np.sqrt(LAMBDA)) @ np.linalg.inv(V) # 根据特征值和特征向量构建矩阵B"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "da0f2576-d262-41cd-8324-208cf3df1c82",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"matrix([[ 1.06066017, -0.35355339],\n",
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" [-0.35355339, 1.06066017]])"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"B"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d70a4893-0524-47aa-91ad-4ad9863a5c89",
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"metadata": {},
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"source": [
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"## 重构矩阵A并打印"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "d07c6472-787a-44c2-9d20-99e4d070826a",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[ 1.25 -0.75]\n",
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" [-0.75 1.25]]\n"
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]
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}
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],
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"source": [
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"A_reproduced = B @ B.T # 通过矩阵B的转置乘积重构矩阵A\n",
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"print(A_reproduced) # 输出重构后的矩阵A"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "85a80909-2aac-49ed-bb7a-f8cc6b80ee7d",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "ecd322f4-f919-4be2-adc3-69d28ef25e69",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.12.7"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}

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